Group theoretic expressions of optical singularities in photonic crystals

Fundamental theoretical insights into the fine structure of electromagnetic fields in photonic crystals are developed, by examining the singular nature of field representations in linear systems, and fundamental design paradigms are established in this meta-material system. Photonic crystals are optical meta-materials that permit the transport of electromagnetic energy that can be tailored by modifying the underlying periodically structured dielectric profile. Propagation is characterized by the dispersion surface through the Bloch states, where frequency wave vector relations characterize system response. A renewed appreciation for the richness of the dispersion relations has led to a deeper consideration of the modal structure itself and its inherent relationship to symmetry. This has led to a group theoretic treatment of the fundamental system space symmetries expressed through local representations of the singular character of the Bloch mode and its vortex states.;Further research on the vectorial character of the local electromagnetic field has uncovered the rich nature of polarization singularities in these periodic microstructures. A group theory representation is used to express its complementarity polarization singularity representation, where fundamental transformation operations permitted by the system's space group are quantified. As a result, the entire electromagnetic field may be determined from the fundamental domain of the system's space group, using derived transformation properties of the local state of polarization. Furthermore, it is shown that local symmetry requires the electromagnetic field to become singular at particular points, known as Wyckoff positions. The reduction of the electromagnetic field to a fundamental domain, the location of the optical singularities, and the transformation properties of the local state of the electromagnetic field allow one to determine the major features of the sub-wavelength structure of the electromagnetic field from fundamental symmetry principles.;In each the case, these fundamental principles were applied to Bloch modes derived in the vanishing dielectric contrast limit. Additional confirmation of theoretical predictions is supported by simulations of high-dielectric-contrast, purely two-dimensional, photonic crystal Bloch modes, in which Maxwell's equations are solved directly using the Finite Element Method (FEM). Finally, studies of optical vortices and polarization singularities within dimensionally confined photonic crystal slabs are studied with three-dimensional FEM simulations, in order to inform the design and fabrication of such structures for future experimental confirmation of the phenomena and possible photonic crystal optical singularity applications.;Central to this study is an analysis of the local energy transport, which has uncovered the existence of optical vortices centered about phase singularities. Further investigation into the energy transport properties at the local level (i.e., far below the scale of the wavelength) reveals optical features never before explicated in the photonic crystal community. It is shown that the electromagnetic mode structure bears the hallmarks of singular optics, whereby the field becomes characterized by singularities---that is, spatial locations where mathematical descriptions of optical properties become undefined. Optical vortices, revealed by energy circulation about a singular point, are expressed by global system space symmetries and the particular character of the dispersion surface. Vortices are the local field magnitude response to their associated phase singularities. In this study, the locations of optical vortices in real space are determined using phasor geometry, and the symmetry rules for their existence are established with group theory. They are categorized into symmetry and accidental singularities, constraining their locations in reciprocal space.